(1) Correlation

• correlation
  • technical details
• random association
  • \(p\) values

correlation:

degree of association or relationship between the observed values taken by two variables (\(X\) and \(Y\))

• Many different ways of doing this (compare group means, regression) are all fundamentally about correlation.
• correlations have a direction:
  • positive: implies that as \(X\) increases, \(Y\) increases
  • negative: \(X\) increases, \(Y\) decreases
• correlations have strength (has nothing to do size of effect):
  • strong: \(X\) and \(Y\) almost always move together
  • weak: \(X\) and \(Y\) do not move together very much
• There is also a technical definition of correlation (later)

What is it?

(Pearson) correlation: also has specific mathematical definition (you don't need to know it):

\[r = \frac{\sum_{i}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i^n(x_i - \bar{x})^2}\sqrt{\sum_i^n (y_i - \bar{y})^2}}\]

This captures extent to which deviations from mean of \(X\) move with deviations from mean of \(Y\).

What is it?

mathematically: correlation is the degree of linear association between \(X\) and \(Y\)

• Takes values between \(-1\) and \(1\)
• Values close to \(1\) or \(-1\) suggest high degree of linear association
• Values close to \(0\) suggest low degree of linear association
• Value of correlation does not tell us how much \(Y\) changes with \(X\)

What is it?

negative correlation: (correlation \(< 0\)) values of \(X\) and \(Y\) move in opposite direction:

• higher values of \(X\) appear with lower values of \(Y\)
• lower values of \(X\) appear with higher values of \(Y\)

positive correlation: (correlation \(> 0\)) values of \(X\) and \(Y\) move in same direction:

• higher values of \(X\) appear with higher values of \(Y\)
• lower values of \(X\) appear with lower values of \(Y\)

• It is possible to see perfect correlation but small change in \(Y\) across \(X\)

• It is possible to see low correlation but large change in \(Y\) across \(X\)

• It is possible to see perfect nonlinear relationship between \(X\) and \(Y\) with \(0\) correlation

weak correlation: values for \(X\) and \(Y\) do not cluster along line

strong correlation: values for \(X\) and \(Y\) cluster strongly along a line

strength of correlation does not fully determine the slope of line describing \(X,Y\) relationship

effect size: this is the slope of the line describing the \(X,Y\) relationship. The larger the effect, the steeper the slope

How do we know a correlation is systematic?

• How do we know that it is not simply a pattern by random chance?
• Apparent patterns can be produced by pure randomness

If you look at enough possible sets of variables, you might find a strong correlation

• But it could have happened by chance!
• So a correlation might not be "real"

(Arbitary Correlations)[http://www.tylervigen.com/spurious-correlations]

Field of statistics investigates properties of chance events (stochastic processes):

• Probability theory tells us how likely events are to happen, given chance
• Can tell us how likely correlation of some value is to happen by chance

How?

1. Compute correlation of \(X\) and \(Y\)
2. How many cases do we have?
   • Patterns with many cases less likely to occur at random
3. Assign a probability that the correlation we see would have happened by chance

ASSUMPTION

• We need to assume we know the chance process generating this correlation

Same Correlation, More cases

statistical significance:

An indication of how likely correlation we observe could have happened purely by chance.

higher degree of statistical significance indicates correlation is less likely to have happened by chance

\(p\) value:

• A numerical measure of statistical significance. Puts a number on how likely observed correlation would have occurred by chance, assuming a we know the chance procedure and the truth is a \(0\) correlation.

• It is a probability, so is between \(0\) and \(1\).

• Lower \(p\)-values indicate greater statistical significance

\(p < 0.05\) often used as threshold for "significant" result.

• but it is not a magic number
• Can observe \(p < 0.05\) by chance (\(\frac{1}{20}\))

\(p\) value:

Be wary of "\(p\)-hacking"

• \(p\) values become meaningless if we look at many associations, then only report the ones that are "significant".

Why?

• low \(p\)-values occur by chance when we look at lots of associations

Recap:

1. Correlations can appear by chance
2. We can assess probability of chance correlation if we know:
   • strength of correlation (close to \(1,-1\))
   • size of the sample (\(N\))
   • underlying chance process
3. \(p\)-values:
   • Obtained using mathematical formulaa

Recap: